A curve has equation 
y = f(x).
(a)
Write an expression for the slope of the secant line through the points 
P(8, f(8))
 and 
Q(x, f(x)).

The slope of the secant line through the points \( P(8, f(8)) \) and \( Q(x, f(x)) \) can be found using the formula for the slope between two points, which is:

\[
\text{slope of the secant line} = \frac{f(x) - f(8)}{x - 8}
\]

This expression represents the slope of the secant line between the two points \( P(8, f(8)) \) and \( Q(x, f(x)) \).

Would you like to go over further details or clarify any concepts?

Here are five follow-up questions related to this problem:

1. What is the definition of a secant line in the context of calculus?
2. How would you find the slope of the tangent line at \( x = 8 \)?
3. What is the relationship between the secant line and the difference quotient?
4. How does the secant line approach the tangent line as \( x \) approaches 8?
5. How do you apply limits to convert the secant slope into the derivative?

**Tip:** The slope of the secant line becomes the slope of the tangent line as the two points get infinitely close, i.e., as \( x \to 8 \). This is the fundamental concept behind derivatives.

A curve has equation y = f(x). (a) Write an expression for the slope of the secant line through the points P(8, f(8)) and Q(x, f(x)).

Discover how to write an expression for the slope of the secant line through points P(8, f(8)) and Q(x, f(x)) on a curve. This detailed solution covers key calculus concepts like secant lines, difference quotients, and limits, making it suitable for high school students studying Pre-Calculus or Calculus.

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